The SI second

The SI second is (currently)
defined as the duration of 9192631770 periods of the radiation
corresponding to the transition between the two hyperfine levels of
the ground state of the cesium 133 atom (implicitly: in the
absence of any external magnetic field). This means the practical
realization of the second is left to atomic clocks,
mostly either cesium clocks (using precisely the definition above),
rubidium clocks, or hydrogen maser clocks. It is possible that
rubidium would, in fact, be a better choice than cesium (to the point
that it would be worth redefining SI?).

Various other time units are defined from the second. The
SI day is 86400 SI seconds. The Julian year
is 365.25 SI days, that is, 31557600
SI seconds. The Gregorian year (rarely used in
astronomy, but it is the mean duration of a year of our calendar) is 365.2425 SI days, that
is, 31556952 SI seconds. Incidentally, the so-called
“light year” (a unit of distance, not time) is the length
of the path traveled by light in vacuum in one Julian year,
which is therefore (taking into account the definition of the SI meter) exactly 9460730472580800
meters. Naturally, the Julian and Gregorian centuries are
one hundred Julian and Gregorian years respectively.

The definition of an SI second prior to 1967 was
related to the Earth's rotation around the sun. Precisely: it is
the fraction 1/31556925.9747 of the tropical year for 1900 January
0 at 12h ephemeris time (the
“ephemeris second”). This definition is probably rather
obscure. Of course, the event 1900 January 0 at 12h ephemeris
time refers to December 31, 1899 at noon; this epoch (ephemeris Julian date 2415020.0) is generally referred to as
“J1900.0”. (The epoch one Julian century later, or 1
January 2000 at 12h ephemeris time (ephemeris Julian date 2451545.0),
is referred to as “J2000.0”.) The expression
the tropical year at (such a date) means that the second is
defined so that the mean geometric longitude of the sun (in other
words, its geometric longitude with all periodic perturbative terms
canceled out) varies by 129602768.13 seconds of arc per Julian century
at the point considered (the former figure is exact according to
CGPM 1960, the latter according to the International
Astronomers' Union 1952, but they differ by a term too small to be
measurable).

Between 1955 and 1958, a joint experiment of the National Physical
Laboratory (UK, where the first cesium frequency samples
had be obtained) and the US Naval Observatory was
conducted to measure the value of the cesium 133 atom's hyperfine
transition frequency (in the then-standard ephemeris second). It is
in this experiment that the value of 9192631770 transitions per second
was measured, which was adopted in 1967 by the Conférence
Générale des Poids et Mesures as the new definition of
the second (keeping the value as close as possible to the former one,
naturally).

Before even the ephemeris second was defined, the second (not
“SI” because this is before SI)
was defined as the fraction 1/86400 of the mean solar day (this is,
after all, the historical definition of the second, the day being
divided in 24 hours of 60 minutes of 60 seconds each). But
aberrations in the Earth's rotation made this definition inherently
inexact. As of J2000, the mean solar day is some 3 milliseconds
longer than the SI day (this represents the secular
increase of the length of the day since about 1820).

The definition of the ephemeris second in terms of the tropical
year 1900 was based on the ephemerides and planetary theories
published by Newcomb at the end of the 19th century in England. This
explains the choice of 1900 as epoch in that definition. Modern
planetary theories tell us that the atomic second adopted in 1967 is
in fact slightly longer than the ephemeris second: it is roughly 6
parts per billion longer. (XXX — check this carefully; this
seems a bit much: perhaps I am wrong in my reading of said planetary
theory.) This also explains why it is around 1820 that the mean solar
day was 86400 seconds long: it is because Newcomb had used data that
spanned, mostly, the 18th and 19th centuries, in doing his
computations (and since the ephemeris second was later adopted as
SI second, it is with this portion of the 1820 day that
we are now left).

Julian date

The Julian date counts days in a certain time standard, usually
either Universal Time Coordinated or Ephemeris Time. On 2000-01-01 at 12:00:00 (noon),
that is, J2000.0, the value of the Julian date was 2451545 exactly.
The modified Julian date is the Julian date minus 2400000.5 (so on
2000-01-01 at 00:00:00 (midnight), or 43200 seconds before J2000.0, it
was 51544).

This has the advantage of defining a running date that does not
have all the bizarre idiosyncrasies of our calendar.

The calendar

The Gregorian calendar

The Gregorian calendar is currently our official calendar for just about the entire
world. Since no standards body has the necessary authority to alter
it (consider, for example, the failed attempt by the French
revolutionaries), it is a safe bet that this calendar will remain
unmodified for a substantial duration.

The calendar year is divided in 365 or 366 days (of 86400
SI seconds): those with 366 days are called leap
years, others are called regular. These days are grouped in
twelve months having the following number of days: 31
(January), 28 or 29 (February: 28 for regular years and 29 for leap
years), 31 (March), 30 (April), 31 (May), 30 (June), 31 (July), 31
(August), 30 (September), 31 (October), 30 (November) and 31
(December). Months are given numbers going from 1 to 12. Days within
a month are given numbers going from 1 to 31 (or less). A calendar
day starts at midnight, when the modified Julian
date is an integer (thus, each calendar day can be associated
precisely one modified Julian date). Years are given (“Common
Era”) numbers consecutively so that the year (of the Gregorian
calendar) that starts on modified Julian date 51544 is given the
number 2000. The rules to determine whether a year is leap are as
follows, for the Gregorian calendar: years whose number is divisible
by 400 (for example, 2000) are leap years; years whose number is
divisible by 100 but not by 400 (for example, 1900) are regular years;
years whose number is divisible by 4 but not by 100 (for example,
2004) are leap years; years whose number is not divisible by 4 are
regular years. Thus, a period of 400 calendar years contains 97 leap
years and 303 regular years, which makes 146097 days in total (or
12622780800 seconds). This is the Gregorian calendar's cycle, and a
fraction 1/400 of this duration is one (average) Gregorian year, or
365.2425 days.

(For what it's worth, the duration of the tropical year in J2000 is
actually 365.24219 SI days, or 365.24218 mean solar days
at that date.)

Calendar days are further grouped in intervals of 7 days (604800
seconds), called weeks. These are named: Monday, Tuesday,
Wednesday, Thursday, Friday, Saturday and Sunday, in this order. The
day of modified Julian date 51544 is a Saturday (the rest can be
deduced from this). Notice that a Gregorian calendar cycle, or 146097
days, contains exactly 20871 weeks. This means that week days will
repeat themselves after 400 calendar years; but since 400 is not
divisible by 7, it implies that the days are not evenly distributed
among the calendar dates (for example, it is a well-known trivia fact
that dates which fall on the 13th of a month are more frequently a
Friday than any other day of the week).

The Julian calendar

The Gregorian calendar is the one we use now. An earlier calendar
is the Julian calendar. This differs from the Gregorian calendar only
in the procedure for computing leap years: a year with a certain
number is a leap year in the Julian calendar if and only if the number
is divisible by 4. The day with modified Julian date 51544,
2000-01-01 (January 1, 2000) on the Gregorian calendar, is 1999-12-19
(December 19, 1999) on the Julian calendar. Week days are otherwise
unaffected: this day is a Saturday for both calendars. The short
cycle of the Julian calendar (that is, without counting week days) is
4 years, or 1461 days, or 126230400 seconds. The long cycle of the
Julian calendar is 28 years, or 1461 weeks, or 10227 days, or
883612800 seconds. It is worth insisting that the Gregorian calendar
only has one cycle of 400 years, with unevenly distributed week days,
whereas the Julian calendar has a small and large cycle, and its week
days are consequently evenly distributed.

Official calendar

The Gregorian calendar was adopted over the Julian calendar at
various times for various countries. Spain made the change as early
as 1582, when the calendar was ordained by pope Gregory XIII,
with Thursday 1582-10-04 on the Julian calendar (MJD
-100841) being followed by Friday 1582-10-15 on the Gregorian calendar
(MJD -100840). England made the change in 1752, and
Wednesday 1752-09-02 (MJD −38780) was followed by
Thursday 1752-09-14 (MJD −38779). Russia made the change
with the revolution, in 1917. Other countries have more complicated
stories (the year 1712 in Sweden was a “double leap year”
because an error in trying to switch from Julian to Gregorian had left
a somewhat confused situation).

Dates in antiquity, “before the Common Era”, that is,
before 0001-01-01 on the Julian calendar, are customarily counted
backwards starting from ~1 (year 1 before the Common Era, which is the
same as year 0 of the Common Era, but the latter is not used;
similarly, year ~2, or 2 before the Common Era, is year −1 of the
Common Era). As far as months and days go, there is nothing to
prevent us from extending the Julian calendar (or, for that matter,
the Gregorian calendar) arbitrarily far back in the past. But what we
now call the Julian calendar has only been consistently applied since
the year 5 of the Common Era. Indeed, years 4, ~1 and ~5 were not
leap: this is the “Augustan reform”, taken by Cæsar
Augustus in ~8 because the pontiffs had incorrectly inserted one leap
year in three (instead of four) between ~45 and ~9. (In other words:
though the Julian calendar prescribed that ~45, ~41, ~37, …,
~13, ~9, ~5, ~1, 4, 8 and 12 be leap, in fact, ~45, ~42, ~39, ~36,
…, ~12, ~9, 8 and 12 were leap.) Besides, it is evidently
anachronistic to use the name “August” for the month named
in honor of he who enacted said reform! For what it's worth, before
that, the month was called Sextilis. Before the Julian reform, it is
about impossible to date anything; “July” was called
Quinctilis before it was dedicated to C. Julius Cæsar, but
the number of days in a month was very variable, and some years even
had 13 months with Mercedonius being the 13th. Also note the the
tradition of starting the year on January (and numbering from January)
was not kept consistently for dates in the Christian period until the
Renaissance (for pre-Christian times, years were designated in Rome by
the names of the Consuls, and these indeed took their office in
January).

The starting point of the calendar, ~0045-01-01, was Julian date 1704987 (at noon ephemeris time). The
assassination of C. Julius Cæsar on the ides of March of
year 44 before the Common Era took place on Julian date 1705426.

Remark: The terms “Common Era” and
“before the Common Era” are to be preferred over the
earlier terms “Anno Domini” (literally, “in the year
of the Lord”) and “Before Christ”: first, they are
civilization-neutral (the calendar we use is, in any case, completely
arbitrary and absurd); moreover, they are more accurate, since the
Jesus referred to as “Christ” by the Christians was
probably born around ~4 (or year 750 of Rome).

Ephemeris Time and avatars (astronomical time scales)

Things get a bit complicated when it comes to defining an actual
time scale, and not just a unit of time. The reason for this is that,
as Einstein's theory of Special Relativity tells us, what we call
“time” is simply one dimension of a four-dimensional
whole, so its measure depends on the choice of axes, choice which is
physically realized as the choice of a reference frame (observers in
relative motion to one another do not measure the same parameter);
furthermore, General Relativity complicates matters even more by
telling us that the four-dimensional space-time continuum is not even
flat but is bent by gravity, so that definitions of time cannot simply
be extended from one point to another.

All ephemeris time scales are defined to be 32.184 seconds ahead of
atomic time on 1977-01-01 at 00:00:00
TAI. There is no real reason behind this number: it is
merely due to a lack of sufficient communication between astronomers
and physicists.

Ephemeris Time proper

Ephemeris Time (TE) is defined by the Earth's
revolution around the sun. This makes no call to a reference frame
except insofar as periodic terms are to be canceled out (and which
periodic terms are there depends on the choice of frame). It is
useful as a long-term standard, but it is useless for short intervals,
as it cannot be realized for such with a satisfactory precision.

The definition of Ephemeris Time is this: the geometric mean
longitude of the Sun, as measured from the vernal equinox of date, is
given by L0 = 279°41′48.04″ +
129602768.13″T + 1.089″T2, where T is Ephemeris
Time measured in Julian centuries since J1900.0. In other words, the
Ephemeris second is defined so that there are 31556925.9747 seconds in
the tropical year at J1900.0, which is defined as that moment close to
0 January 1900 noon such that the geometric mean longitude of the Sun
was 279°41′48.04″ (this is completely theoretical: in
practice, J1900.0 would be defined by Universal
Time).

This expression is taken from Newcomb's ephemerides, but more
accurate theories and the redefinition of the SI second
in terms of atomic clocks have made Ephemeris Time
proper obsolete.

Barycentric Coordinate Time

Barycentric Coordinate Time, or TCB is the most
fundamental time scale that can be defined. Essentially, it is the
proper time measured by an asymptotic observer (i.e. one located
infinitely far away) at rest with respect to the solar system. More
precisely, it is the time coordinate of a space-time coordinate system
that is centered in space on the barycenter of the solar system, that
is nonrotating with respect to distant galaxies, and that tends
asymptotically to the proper time of an observer at rest with respect
to the coordinate system. It was introduced by the International
Astronomers' Union in 1991, following the need to clarify the
definition of Barycentric Dynamical Time, which is
somewhat dubious in principle.

Barycentric Dynamical Time

Barycentric Dynamical Time, or TDB, is the same as Barycentric Coordinate Time only differing by a
constant factor which is chosen so as to make TDB differ
from Terrestrial Time by only periodic and Poisson
terms. In other words, whereas it is defined by a coordinate system
centered at the barycenter of the solar system, TDB is
actually chosen to agree “in the mean” (up to periodic and
Poisson terms) with a clock located on the Earth's surface.

On the one hand, this is a rather dubious definition. It is not
entirely clear what is a periodic (or Poisson) term and what is a
secular term (especially when the periodic terms in question are of
very long period); this is the reason why the IAU in 1991
decided to recommend the use of TCB
over TDB. On the other hand, TDB is very
useful because it actually models the Ephemeris
Time used in older planetary theories (indeed, planetary theories
are calculated in a barycentric coordinate system, and the actual
second length was historically that measured on Earth) — such as
those of Newcomb. To avoid breaking continuity with earlier works,
TDB is still used in constructing planetary theories.

Effectively, the ratio of TCB to TDB is 1
+ 1.55051979176×10−8. This constant reflects, approximately,
for two thirds the gravitational redshift caused by the Sun's
attraction, for one third the time dilation produced by the Earth's
movement around the Sun, for around 5% the gravitational redshift
caused by the Earth's own attraction, and for yet smaller parts other
effects such as the other planets' influence (all these effects are
absent in TCB but present in TDB). A
different way of stating this is that whereas TCB
measures the time seen by an asymptotic observer in SI
seconds, on the other hand TDB measures it in
TDB seconds, where a TDB second is equal to
an SI second on Earth but made longer by 15.5 parts in a
billion because of the two effects mentioned (similarly, the
TDB meter is longer than the SI meter in the
same proportion, since the speed of light is kept constant).

The two scales are synchronized by stating that they coincide on
1977-01-01 at 00:00:00 TAI. It
follows that since then TCB has run ahead of
TDB by a dozen seconds.

Geocentric Coordinate Time

Geocentric Coordinate Time, or TCG, is the same as Barycentric Coordinate Time, except that it uses the
Earth's center, rather than the barycenter of the solar system (and an
“asymptotic” observer is one free from the gravitational
field of the Earth rather than that of all the solar system). Thus,
TCG coincides with Terrestrial Time
except insofar as it omits the gravitational effect of the Earth
itself (and the time dilation caused by its rotation, but these are
very small indeed). The ratio of TCG to TT
is 1 + 6.969290139×10−10. The scales are synchronized in the
same way that TCB and TDB are.

The relation of TCG to TCB is truly the
scope of General Relativity. In particular, note that their relation
depends on the point in space where the comparison is made:
this is because a terrestrial observer and one at the center of the
solar system might not agree about whether two events are
instantaneous. The relation of TCG to TCB
has three kinds of terms:

A secular term: the ratio of TCB to TCG
tends to 1 + 1.48082688934×10−8 at infinity. This constant
reflects, approximately, for two thirds the gravitational redshift
caused by the Sun's attraction, for one third the time dilation
produced by the Earth's movement around the Sun, and for yet smaller
parts other effects such as the other planets' influence: these
effects are present in TCG but not in TCB.
Note that the gravitational effect of the Earth itself is, as we have
said, not present in TCG, whereas it is in
TT.

Periodic terms: these can amount up to slightly less than 2
milliseconds. The leading periodic term has period one year and
amplitude 1657 microseconds; it is currently maximal in the beginning
of April (making TCB ahead of TCG by about
1.7 milliseconds more than in the beginning of January: this is due to
the ellipticity of the Earth's orbit).

A term dependent on the observer's position, which cancels if the
observer is at the center of the Earth (it just cancels by convention,
in the sense that the terms at the center of the Earth are the terms
already mentioned above). This term is at most of the order of 2
microseconds for an observer on the Earth's surface (TCB
is ahead of TCG by about 2 microseconds more for an
observer situated on the equator, at sunrise, on the summer or winter
solstice, than for the center of the Earth, because such an observer
is ahead of the Earth's revolution).

Terrestrial Time

Terrestrial Time, or TT (sometime called Terrestrial
Dynamical Time, TDT), is, as its name indicates, the time
scale on Earth. It is the proper time measured on the surface of the
(rotating) geoid. Like all time scales defined in this section, but unlike Atomic
Time, Terrestrial Time is an ideal time scale: in other words, it
extends arbitrarily far in the past and the future (well, in fact, it
doesn't: it only extends as far as the Earth exists in much the same
form as it now does; if you need to go further than that, you need barycentric time). Its actual realization is
based on Atomic Time: precisely, TT is
32.184 seconds ahead of TAI, to the extend that
TAI can be trusted.

TT differs from TDB by
the same periodic and local terms as were mentioned for the difference
of TCG from TCB (see under TCG); but there is no secular term, by
definition of TDB.

International Atomic Time

International Atomic Time, or TAI, is a practical
realization of Terrestrial Time (TT is
an abstract scale, whereas TAI is a measured time;
further, they differ by that annoying 32.184 seconds). Contrary to Coordinated Universal Time, TAI is a
linear standard: to the best precision achieved, it ticks by one
second every SI second.

TAI was chosen to coincide with the Universal Time UT1 at midnight on 1958-01-01 (modified
Julian date 36204): this is the start of TAI, and it
should be emphasized that it makes no sense to refer to a date prior
to this in TAI (instead, TT should be used, which can be realized
approximately by Ephemeris Time if
astronomical data are available). (Actually, TAI has
been available since July 1955 or so, but the epoch had not yet been
fixed.) TAI received the blessings of the 14th
Conférence Générale des Poids et
Mesures in 1971.

TAI is the master time standard: not only does it realize
TT, but it also serves to define UTC, and, consequently, legal time; it is
also used to define certain time standards such as GPS
time.

TAI is maintained by the Bureau International des Poids et
Mesures which periodically averages various atomic clocks around
the world to compute TAI, and publishes the
offsets of the various clocks to TAI (the master clock of
the Observatoire de Paris is used to centralize the offset
computation). The unpleasant consequence of this is that, if very
high precision is required, TAI can only be known a
posteriori, after the offsets have been published.

Coordinated Universal Time

Coordinated Universal Time, or UTC is the official
standard of time throughout the world, being the basis for legal time.
It is to UTC that time zone offsets (generally an integer
number of hours) are added to produce local time. UTC
has existed since 1961 (prior to this, the basis for legal time was Universal Time, not coordinated to Atomic Time).

UTC does not flow linearly: it is defined with respect
to TAI, and is maintained to within
0.9 seconds of UT1. Since 1972,
UTC has been defined as offset from TAI by
an integer number of seconds. Currently, this offset is 33 (that is,
TAI is exactly 33 seconds ahead of UTC):
this is true since 2006-01-01 and will remain true at least until
2006-06-30.

The difference between TAI and UTC (since
1972) is increased (or possibly decreased) in the form of leap
seconds, during which UTC moves from 23:59:59 to 23:59:60
before becoming 00:00:00 (or skips 23:59:59 if a negative leap second
is ordained). Leap seconds can be inserted before January 1 or July 1
on a given year (or theoretically April 1 or October 1); negative leap
seconds have never been used.

The decision on when to insert a leap second is made by the International Earth Rotation
Service (central bureau, now in Frankfurt), based on observations
of the Earth's orientation in space, and is published in the
IERSBulletin
C. As of writing, the most
recent Bulletin C published the decision to insert a leap second
at the end of December 2005, thus increasing the
TAI−UTC difference from the value 32 (that
it has had since January 1, 1999) to 33.

The atomic clocks used to define TAI also compute a
local version of UTC. This is, in fact, the time
standard generally used to define legal time in the country in
question. For example, legal time in the United States of America is
defined by adding the time zone offset to the UTC reading
of the US Naval
Observatory's Master Clock; the
difference between UTC(USNO) and
UTC(BIPM) is typically of the order of a few
nanoseconds (and is, of course, kept as small as possible). The
reason why the BIPM's definition of UTC
cannot be used directly as a basis for legal time is that it is known
only a posteriori, when the various atomic clocks have been
read.

Before 1972, the relation between UTC and
TAI was more complex, to make UTC agree with
Universal Time more closely. Not only were jump
discontinuities used occasionally (e.g. a 50 millisecond
discontinuity on 1961-08-01), but also the clock frequency (the rate
of flow of UTC) was slightly skewed: the function giving
UTC from TAI was piecewise affine, and
UTC did not always tick by one second every
SI second, even after the second was defined in terms of
atomic clocks in 1967.

The historical
table of TAI−UTC differences is
available for precise computations of durations, even for dates prior
to 1972 (most tables list only the leap seconds, the latter part of
the table). The same information is available in a slightly different
format from the IERS.

The Time Service Department of the U.S. Naval Observatory issued a
questionnaire
on December 1999 with the idea of possibly completely doing away with
leap seconds. More has not been heard from this.

Universal Time (GMT, UT0,
UT1, UT2)

Universal Time has for basis the rotation of the Earth. The mean
solar time on Greenwich meridian is called GMT, for
“Greenwich Mean Time”. (Actually, historically, Greenwich
Mean Time was measured from mid-day, so that it was twelve hours
behind Greenwich civil time. Nowadays, the term is imprecise and
should no longer be used.)

The observed mean sidereal time at Greenwich (measured from the
position of certain reference stars) is called GMST; it
is transformed to give a mean solar time, the first Universal Time
standard, or UT0. This in turn gives rise to a time
standard, UT1, that corrects for nutation and pole
movement. Two further periodic terms (with periods one year and half
a year) are removed from UT1 to form UT2.
There remains some effects due to tidal forces that slow the Earth's
rotation.

Universal Time is rather imprecise, because the rotation of the
Earth is subject to so many perturbative effects. The relations
between UT1 and UT0, and between
UT2 and UT1, are conventional (the latter is
simple: UT2−UT1 is 0.022 sin(2πT) − 0.012
cos(2πT) − 0.006 sin(4πT) + 0.007 cos(4πT) where T is the
time in Besselian years of 365.2422 days; for the former, consult
Aoki, Guinot, Kaplan, Kinoshita, McCarthy & Seidelmann,
Astron. Astrophys., 105 (1982)).

Not only is the flow of UT1 irregular, but its mean
rate is not even unity: it increases by one day, on the mean, once
every mean solar day, and the mean solar day is presently somewhere
around 3 milliseconds longer than the SI day — this
is why leap seconds are added quite regularly to UTC (and
negative leap seconds will probably never happen).

SI units

The SI (Système International), comprising the
meter, the kilogram, the second, the ampere, the kelvin and the
candela as base units, was named by the 11th Conférence
Générale des Poids et Mesures in 1960. The 14th
Conférence Générale des Poids et
Mesures added the mole as a new base unit in 1971.

The metre is the length of the path traveled by light in vacuum
during a time interval of 1/299792458 of a second. (17th
Conférence Générale des Poids et
Mesures, 1983)

(Former definition: The metre is the length equal to 1650763.73
wavelengths in vacuum of the radiation corresponding to the transition
between the levels 2p10 and 5d5 of the
krypton 86 atom. (11th Conférence
Générale des Poids et Mesures, 1960))

(Former definition: The unit of length is the metre, defined by
the distance, at 0° [Celsius], between the axes of the two central
lines marked on the bar of platinum-iridium kept at the Bureau
International des Poids et Mesures and declared Prototype of the metre
by the 1st Conférence Générale des Poids et
Mesures, this bar being subject to standard atmospheric
pressure and supported on two cylinders of at least one centimetre
diameter, symmetrically placed in the same horizontal plane at a
distance of 571mm from each other. (7th Conférence
Générale des Poids et Mesures, 1927))

The kilogram is the unit of mass; it is equal to the mass of the
international prototype of the kilogram. (1st
Conférence Générale des Poids et
Mesures, 1889 & 3rd Conférence
Générale des Poids et Mesures, 1901) This
prototype is made of an alloy of platinum with 10 per cent iridium in
mass (to within 0.0001). It was manufactured in the late 1880's (as a
copy of the prior kilogram prototype kept at the French Archives). It
is kept in a vault at the the Bureau International des Poids et
Mesures in Sèvres, France.

The second is the duration of 9192631770 periods of the
radiation corresponding to the transition between the two hyperfine
levels of the ground state of the cæsium 133 atom.
(13th Conférence Générale des Poids et
Mesures, 1967)

(Former definition: The second is the fraction 1/31556925.9747
of the tropical year for 1900 January 0 at 12h ephemeris time
(Comité International des Poids et Mesures, 1956,
ratified by 11th Conférence Générale des
Poids et Mesures, 1960))

The ampere is that constant current which, if maintained in two
straight parallel conductors of infinite length, of negligible
circular cross-section, and placed one metre apart in vacuum, would
produce between these conductors a force equal to 2×10−7
[newton] per metre of length. (Comité International
des Poids et Mesures, 1946 & 9th Conférence
Générale des Poids et Mesures, 1948)

The kelvin, unit of thermodynamic temperature, is the fraction
1/273.16 of the thermodynamic temperature of the triple point of
water. (10th Conférence Générale des
Poids et Mesures, 1954, made explicit by 13th
Conférence Générale des Poids et
Mesures, 1967)

The candela is the luminous intensity, in a given direction, of
a source that emits monochromatic radiation of frequency
540*1012 hertz and that has a radiant intensity in that
direction of 1/683 watt per steradian. (16th
Conférence Générale des Poids et
Mesures, 1979)

(Former definition: The candela is the luminous intensity, in
the perpendicular direction, of a surface of 1/600000 square metre of
a black body at the temperature of freezing platinum under a pressure
of 101325 newtons per square metre. (Comité
International des Poids et Mesures, 1946, ratified by 9th
Conférence Générale des Poids et
Mesures, 1948, explicited by 13th Conférence
Générale des Poids et Mesures, 1967))

The mole is the amount of substance of a system which contains
as many elementary entities as there are atoms in 0.012 kilogram
of carbon 12; its symbol is “mol”. When the mole is
used, the elementary entities must be specified, and may be atoms,
molecules, ions, electrons, other particles, or specified groups of
such particles. (14th Conférence
Générale des Poids et Mesures, 1971)

The Système International is used throughout the world. The
United States of America (together with Liberia and Burma) are a
notable exception: their system of unit is derived from the
SI so as to keep values very close to the former British
Imperial system. Thus, the yard was defined in 1959 by the
US National Bureau of Standards as 0.9144 meters (with
the notable exception of the US survey foot, which is
1200/3937 meters, a value derived from an earlier identification) and
the pound as 0.45359237 kilograms (as for the gallon, it is 231 cubic
inches, or 3.785411784 liters exactly). Note that a temperature in
degrees Celsius is obtained by taking the (absolute thermodynamic)
temperature in Kelvins and subtracting 273.15; the temperature in
degrees Fahrenheit is obtained as an affine function of the temperature
in degrees Celsius: −40 Fahrenheit is −40 Celsius, and the slope is
such that 9 degrees Fahrenheit are 5 degrees Celsius.